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Mirrors > Home > MPE Home > Th. List > tpostpos2 | Structured version Visualization version GIF version |
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
tpostpos2 | ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpostpos 7941 | . 2 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | |
2 | relrelss 6105 | . . . 4 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V)) | |
3 | ssun1 4062 | . . . . . 6 ⊢ (V × V) ⊆ ((V × V) ∪ {∅}) | |
4 | xpss1 5544 | . . . . . 6 ⊢ ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V) |
6 | sstr 3885 | . . . . 5 ⊢ ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) | |
7 | 5, 6 | mpan2 691 | . . . 4 ⊢ (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
8 | 2, 7 | sylbi 220 | . . 3 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
9 | df-ss 3860 | . . 3 ⊢ (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) | |
10 | 8, 9 | sylib 221 | . 2 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) |
11 | 1, 10 | syl5eq 2785 | 1 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 Vcvv 3398 ∪ cun 3841 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 {csn 4516 × cxp 5523 dom cdm 5525 Rel wrel 5530 tpos ctpos 7920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-fv 6347 df-tpos 7921 |
This theorem is referenced by: 2oppchomf 17098 mattpostpos 21205 |
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